• SOME STUDIES ON ZERO DIVISOR GRAPHS ASSOCIATED WITH CONNECTED RINGS
Abstract
Ashrafi studies the properties of the unit graph associated with rings. In this paper we present the properties of the unit graph of a connected ring. If R is a connected ring and U (R)is the set of unit elements of R, then the unit graph of R denoted by G (R) is the graph obtained by setting all the elements of R to the vertices and defining distinct vertices x and y to be adjacent if and only if x+y Î U(R). We prove that if R is a finite ring, then the following statements hold for the unit graph of R. (a) If 2 Ï U(R) then the unit graph G (R) is a |U(R)|- regular graph (b) If 2Î U(R), then for every x Î U(R) we have deg (x) = |U(R)|-1 and for every x ÎR/U(R) we have deg (x)= |U(R)| .
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